Vibratory rotation sensor with whole-angle tracking

ABSTRACT

The invention is a vibratory rotation sensor comprising a resonator and a housing to which the resonator is attached and a method for reading out the standing-wave orientation angle utilizing a tracking angle which is maintained equal to the orientation angle on average. The resonator is a rotationally-symmetric thin-walled object that can be made to vibrate in a plurality of standing-wave modes. The method includes applying driving voltages to housing electrodes and determining the orientation of a standing wave by performing operations on the resonator signal that arrives at a single resonator output port from one or more electrodes in close proximity to the housing electrodes. A driving voltage may include either a pair of excitation voltages or a forcing voltage or both. An excitation voltage has essentially no effect on the resonator dynamics but carries information pertaining to the tracking angle and the standing-wave parameters when it arrives at the resonator output port. A forcing voltage causes forces to be applied to the resonator and thereby affects the dynamics of the resonator and the standing-wave parameters. The driving voltages applied to the housing electrodes are brought together into a single resonator signal as a result of being transmitted through the housing-electrode-resonator-electrode capacitances to the resonator output port. In order to extract the standing-wave orientation angle, the excitation and forcing voltages are designed to be separable by appropriate operations performed on the resonator signal.

CROSS-REFERENCES TO RELATED APPLICATIONS

The subject matter of this invention is shared by the inventions disclosed in patent applications Vibratory Rotation Sensor with AC Forcing and Sensing Electronics by Kumar and Foster, Vibratory Rotation Sensor with Multiplex Electronics (U.S. patent application Ser. No. 08/802,006) by Matthews, Darling, and Varty, and Vibratory Rotation Sensor with AC Forcing Voltages by Lynch (U.S. patent application Ser. No. 08/802,007).

CROSS-REFERENCES TO RELATED APPLICATIONS

The subject matter of this invention is shared by the inventions disclosed in patent applications Vibratory Rotation Sensor with AC Forcing and Sensing Electronics by Kumar and Foster, Vibratory Rotation Sensor with Multiplex Electronics (U.S. patent application Ser. No. 08/802,006) by Matthews, Darling, and Varty, and Vibratory Rotation Sensor with AC Forcing Voltages by Lynch (U.S. patent application Ser. No. 08/802,007).

BACKGROUND OF THE INVENTION

This invention relates generally to vibratory rotation sensors and more specifically to the electronics associated with such rotation sensors.

A prior-art vibratory rotation sensor (VRS) 10 consisting of an outer member 12, a hemispherical resonator 14, and an inner member 16, all made of fused quartz and joined together with indium, is shown unassembled in FIG. 1. The inertially-sensitive element is the thin-walled, 5.8-cm-diameter hemispherical resonator 14 positioned between the outer member 12 and the inner member 16 and supported by the stem 26.

A ring forcer electrode 20 and sixteen discrete forcer electrodes 22 are deposited on the interior surface of the outer member 12. In the assembled VRS 10, the ring forcer electrode 20 and the sixteen discrete forcer electrodes 22 are in close proximity to the exterior metalized surface 32 of the hemispherical resonator 14. In the assembled VRS, eight pickoff electrodes 24 deposited on the inner member 16 are in close proximity to the interior metalized surface 30 of the hemispherical resonator 14.

Capacitive forces can be exerted on the hemispherical resonator 14 by means of appropriate forcing voltage differences between the hemispherical resonator 14 and the ring forcer electrode 20 to cause the hemispherical resonator to vibrate in the lowest-order inextensional (or flexing) mode. A standing wave is established having four antinodes at 90-degree intervals about the circumference with four nodes offset by 45 degrees from the antinodes. The 0-degree and 180-degree antinodal points oscillate 90 degrees out of phase with 90-degree and the 270-degree antinodal points. The standing wave causes the shape of the rim of the hemispherical resonator to change from circular to elliptical (with semi-major axis through the 0-degree/180-degree antinodes) to circular to elliptical (with semi-major axis through the 90-degree/270-degree antinodes.

Rotation of the VRS 10 about an axis normal to the plane of the hemispherical-resonator rim 34 causes the standing wave to rotate in the opposite direction with respect to the VRS by an angle proportional to the angle of rotation of the VRS 10. Thus, by measuring the angle of rotation of the standing wave with respect to the VRS 10, one can determine the angle of rotation of the VRS 10.

The vibrational mode of the hemispherical resonator 14 is excited by placing a DC bias voltage on the hemispherical resonator 14 and an AC voltage on the ring forcer electrode 20, the frequency of the AC voltage being twice the resonant frequency of the hemispherical resonator 14.

The standing-wave pattern angle with respect to the VRS 10 is determined by measuring the currents that flow into and out of the pickoff electrodes 24 as the hemispherical resonator 14 vibrates and the capacitances of the pickoff electrodes 24 with respect to the hemispherical resonator vary. An x axis signal I_(x) is obtained from the combination I₀ -I₉₀ +I₁₈₀ -I₂₇₀ where the subscripts identify the angular orientations relative to the x axis of the electrodes from which the currents originate. Similarly, a y axis signal I_(y) is obtained from the combination I₄₅ -I₁₃₅ +I₂₂₅ -I₃₁₅. The tangent of twice the standing-wave pattern angle with respect to the 0-degree (i.e. x) axis is given by the ratio of I_(y) to I_(x).

As a result of nonuniformities in the thickness of the hemispherical resonator 14, the establishment of a first standing wave will lead to the development of a second standing wave oscillating in phase quadrature with antinodes coinciding with the nodes of the first standing wave. The development of a second standing wave can be inhibited by placing appropriate voltages on the sixteen discrete forcer electrodes 22.

A DC bias voltage is typically maintained on the hemispherical resonator 14 in order to reduce the magnitudes of the AC forcing voltages applied to the ring forcer electrode 20 and the discrete forcer electrodes 22 and to make the force exerted on the resonator a linear function of the AC driving voltages. The presence of the DC bias voltage results in slow changes in the electrical properties of the VRS which have been attributed to capacitance changes caused by charge-migration phenomena taking place at or within the outer member 12 and the inner member 16. These slow changes have resulted in an unacceptably large performance degradation over time and special means have to be provided to compensate for these effects.

SUMMARY OF THE INVENTION

The invention is a vibratory rotation sensor comprising a resonator and a housing to which the resonator is attached and a method for reading out the standing-wave orientation angle of such a sensor utilizing a tracking angle which is maintained equal to the orientation angle on average. The resonator is a rotationally-symmetric thin-walled object that can be made to vibrate in a plurality of standing-wave modes. The orientation of a standing wave is specified by the orientation angle of a particular antinodal axis of the standing wave with respect to a fixed point on the resonator.

One or more electrodes are deposited on a surface of the resonator and electrically connected to a single output port. The housing has a plurality of deposited electrodes in close proximity to the one or more resonator electrodes.

The method for reading out the orientation angle for a particular standing-wave includes the steps of generating a plurality of excitation voltages, applying the excitation voltages to the housing electrodes, and determining the difference between the orientation angle and a synthesized tracking angle by performing operations on the resonator signal appearing at the output port of the resonator.

The driving voltages applied to the housing electrodes are communicated to the resonator output port by means of the capacitances that exist between the housing electrodes and the resonator electrodes. A driving voltage includes either a pair of excitation voltages or a forcing voltage or both. An excitation voltage has essentially no effect on the resonator dynamics but carries information pertaining to the tracking angle and the standing-wave parameters when it arrives at the resonator output port. A forcing voltage causes forces to be applied to the resonator and thereby affects the dynamics of the resonator and the standing-wave parameters.

The driving voltages applied to the housing electrodes are brought together into a single resonator signal as a result of being transmitted through the housing-electrode-resonator-electrode capacitances to the resonator output port. In order to determine the difference between the orientation angle of a standing wave and the tracking angle, the excitation and forcing voltages are designed to be separable by appropriate operations performed on the resonator signal.

The excitation and forcing voltages can be structured in a variety of ways. A frequency-division multiplexing approach results in the excitation voltages being confined to separated frequency bands and the frequency spectrum of the forcing voltages being confined to a frequency band separated from the frequency bands associated with the excitation voltages.

A phase-division multiplexing approach results in the excitation voltages being periodic functions of time with the same frequency but with phases differing by a quarter of a cycle, the frequency spectrum of the forcing voltages being confined to a frequency band separated from the frequencies of the excitation voltages.

One time-division multiplexing approach results in the excitation voltages being proportional to unique square waves that take on the values 0 and 1, and each forcing voltage including a multiplicative factor proportional to a square wave that takes on the values 0 and 1 where only one of the square waves associated with the excitation and forcing voltages takes on the value 1 at any given time.

A second time-division multiplexing approach results in each excitation voltage being proportional to the product of a periodic function of time having a predetermined frequency and phase and a unique square wave that takes on the values 0 and 1 and each forcing voltage including a multiplicative factor proportional to a square wave that takes on the values 0 and 1, only one of the square waves associated with the excitation and forcing voltages taking on the value 1 at any given time.

A code-division multiplexing approach results in the excitation voltages being proportional to unique square waves which take on the values of -1 and 1 in accordance with predetermined pseudorandom sequences, the frequency spectrum of the forcing voltages being confined to a frequency band separated from the frequency band associated with the excitation voltages.

The process of determining the orientation angle of a standing-wave from the resonator signal consists of first extracting at least two components from the resonator signal and then determining the difference between the orientation angle and the tracking angle by performing operations on the two components. In the case of frequency-division multiplexing wherein the two components occupy separated frequency bands, each component is extracted by performing operations on the resonator signal that discriminate between the separated frequency bands.

In the case of phase-division multiplexing wherein the two components are periodic functions having the same frequency and phases that differ by one-quarter of a cycle, each component is extracted by performing operations on the resonator signal that discriminate between the phases of the two components.

In the case of time-division multiplexing wherein the two components are present in the resonator signal during different time periods, each component is extracted by performing operations on the resonator signal that discriminate between the different time periods.

In the case of code-division multiplexing wherein the two components are pseudorandom sequences of 0's and 1's and the cross correlation of the pseudorandom sequences is 0, each component is extracted by performing operations on the resonator signal that discriminate between the two pseudorandom sequences.

The tracking angle is continually adjusted so as to maintain the difference between the orientation angle and the tracking angle at zero on average. The orientation angle is calculated by adding the tracking angle to the difference between the orientation angle and the tracking angle.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 shows the structure of a prior-art vibratory rotation sensor.

FIG. 2 shows a block diagram of the control and readout electronics for the invention.

FIG. 3 shows the multiplex control signals for a particular embodiment of the invention.

DESCRIPTION OF THE PREFERRED EMBODIMENTS

The invention is a vibratory rotation sensor for which the control and readout is accomplished with multiplexed signals. The vibratory rotation sensor of the present invention consists of a resonator, a housing to which the resonator is attached, and multiplex electronics. The resonator can be any rotationally-symmetric thin-walled object having standing-wave vibration modes. The prior art typically suggests that the resonator be hemispherical in shape.

A simplified method for determining the parameters of the standing waves and controlling the dynamics of the resonator is illustrated in FIG. 2. The standing waves are describable with respect to x and y axes fixed with respect to the resonator. The orientation of the inphase standing wave with respect to the resonator can be specified by the orientation angleθ of an inphase antinodal axis measured clockwise from the x axis. The deviation of the resonator rim from a circle along the inphase antinodal axis is assumed to vary as cos(ωt+θ) where ω is the vibration frequency, t is time, and φ is an arbitrary phase angle. The orientation of the quadrature standing wave with respect to the resonator is specified by the orientation angledθ+π/4 of a quadrature antinodal axis measured clockwise from the x axis. The deviation of the resonator rim from a circle along the quadrature antinodal axis is assumed to vary as sin(ωt+φ).

The circumferentially-continuous resonator electrode 42, deposited on the interior surface of the resonator, is biased to a DC voltage V_(B) and is connected through a DC-blocking capacitor 43 to the amplifier-demultiplexer 44. Eight electrodes 46 attached to the VRS housing are equally spaced about the circumference in close proximity to the resonator electrode 42, the uppermost xp electrode being centered on the x-axis. The eight electrodes 46 are supplied with the driving voltages V_(xp) (t), V_(xn) (t), V_(yp) (t), and V_(yn) (t) from the multiplexer 48 where

    V.sub.xp (t)=V.sub.mxr (t) cos 2θ.sub.r cos (ω.sub.xr t+ψ.sub.xr)-V.sub.myr (t) sin 2θ.sub.r cos (ω.sub.yr t+ψ.sub.yr)+V.sub.cx (t)U.sub.xp (t)

    V.sub.xn (t)=-V.sub.mxr (t) cos 2θ.sub.r cos (ω.sub.xr t+ψ.sub.xr)+V.sub.myr (t) sin 2θ.sub.r cos (ω.sub.yr t+ψ.sub.yr)+V.sub.cx (t)U.sub.xn (t)

    V.sub.yp (t)=V.sub.mxr (t) sin 2θ.sub.r cos (ω.sub.xr t+ψ.sub.xr)+V.sub.myr (t) cos 2θ.sub.r cos (ω.sub.yr t+ψ.sub.yr)+V.sub.cy (t)U.sub.yp (t)                  (1)

    V.sub.yn (t)=-V.sub.mxr (t) sin 2θ.sub.r cos (ω.sub.xr t+ψ.sub.xr)-V.sub.myr (t) cos 2θ.sub.r cos (ω.sub.yr t+ψ.sub.yr)+V.sub.cy (t)U.sub.yn (t)

The excitation voltages V_(mxr) (t)cos(ω_(xr) t+ψ_(xr)) and V_(myr) (ω_(yr) t+ψ_(yr)) are components in the x_(r) -y_(r) tracking-angle coordinate system of FIG. 2 (denoted by the r in the subscripts). The preferred embodiments of the excitation voltages include the sinusoids cos(ω_(xr) t+ψ_(xr)) and cos(ω_(yr) t+ψ_(yr)). There are a variety of periodic functions F(ω_(xr) t+ψ_(xr)) which may be utilized instead of the sinusoids including ordinary square waves.

The x_(r) -axis is rotated clockwise from the x-axis by the tracking angle θ_(r). The excitation voltages are designed not to affect the parameters of a standing wave on the resonator. The angular frequencies ω_(xr) and ω_(yr) and phases ψ_(xr) and ψ_(yr) depend on the type of multiplexing being used. The forcing voltages V_(cx) (t)U_(xp) (t), V_(cx) (t)U_(xn) (t), V_(cy) (t)U_(yp) (t), and V_(cy) (t)U_(yn) (t) (expressed as components in the x-y coordinate system) cause forces to be applied to the resonator for the purpose of controlling the parameters of the one or more standing waves on the resonator. The functions U_(xp) (t), U_(xn) (t), U_(yp) (t) and U_(yn) (t) are generated by control unit 50 and supplied to multiplexer 48. The voltages V_(cx) (t) and V_(cy) (t) are predetermined functions used to isolate the forcing voltages from the excitation voltages.

The current I(t) flowing from the resonator electrode 42 into the amplifier-demultiplexer 44 is given by

    I(t)=I.sub.xp (t)+I.sub.xn (t)+I.sub.yp (t)+I.sub.yn (t)   (2)

where

    I.sub.xp (t)=K.sub.I  V.sub.mxr (t)ω.sub.xr cos 2θ.sub.r cos (ω.sub.xr t+ψ.sub.xr)-V.sub.myr (t)ω.sub.yr sin 2θ.sub.r cos (ω.sub.yr t+ψ.sub.yr)+V.sub.cx (t)ωUxpU.sub.xp (t)!C.sub.xp

    I.sub.xn (t)=K.sub.I  -V.sub.mxr (t)ω.sub.xr cos 2θ.sub.r cos (ω.sub.xr t+ψ.sub.xr)+V.sub.myr (t)ω.sub.yr sin 2θ.sub.r cos (ω.sub.yr t+ψ.sub.yr)+V.sub.cx (t)ωUxnU.sub.xn (t)!C.sub.xn

    I.sub.yp (t)=K.sub.I  V.sub.mxr (t)ω.sub.xr sin 2θ.sub.r cos (ω.sub.xr t+ψ.sub.xr)+V.sub.myr (t)ω.sub.yr cos 2θ.sub.r cos (ω.sub.yr t+ψ.sub.yr)+V.sub.cy (t)ωUypU.sub.yp (t)!C.sub.yp                        (3)

    I.sub.yn (t)=K.sub.I  -V.sub.mxr (t)ω.sub.xr sin 2θ.sub.r cos (ω.sub.xr t+ψ.sub.xr)-V.sub.myr (t)ω.sub.yr cos 2θ.sub.r cos (ω.sub.yr t+ψ.sub.yr)+V.sub.cy (t)ω.sub.Uyn U.sub.yn (t)!C.sub.yn

The capacitances C_(xp), C_(xn), C_(yp), and C_(yn) are the capacitances of the xp, xn, yp, and yn electrodes 46 with respect to the resonator electrode 42. The angular frequencies ω_(Uxp), ω_(Uxn), ω_(Uyp), and ω_(Uyn) are those associated with the corresponding U's and are typically equal to or less than 2ω where ω is the resonator vibration frequency. The symbol K₁ denotes a constant. The phase differences between the driving voltages and the resulting currents are of no relevance and have been ignored in the equations above. The capacitances are given by

    C.sub.xp =C.sub.o  1+d.sub.i cos 2θ cos (ωt+φ)-d.sub.q sin 2θ sin (ωt+φ)!

    C.sub.xn =C.sub.o  1-d.sub.i cos 2θ cos (ωt+φ)+d.sub.q sin 2θ sin (ωt+φ)!                            (4)

    C.sub.yp =C.sub.o  1+d.sub.i sin 2θ cos (ωt+φ)+d.sub.q cos 2θ sin (ωt+φ)!

    C.sub.yn =C.sub.o  1-d.sub.i sin 2θ cos (ωt+φ)-d.sub.q cos 2θ sin (ωt+φ)!

where terms involving higher orders of d_(i) and d_(q) have been omitted. The effects of the higher-order terms are taken into account in subsequent processing operations. The quantity C_(o) is the capacitance of the electrode pairs when the resonator is not excited, d_(i) and d_(q) are the maximum flexing amplitudes respectively of the inphase and quadrature modes divided by the gap between the resonator electrode 42 and the electrodes 46 when the resonator is not excited, θ is the angle between the antinode of the inphase standing wave and the x-axis, ω is the angular frequency of vibration of the resonator, and φ is an arbitrary phase angle.

Substituting the expressions for the capacitances in the current equations and summing to obtain I, we obtain ##EQU1## The current I(t) is transformed into the voltage V(t) by the amplifier-demultiplexer 44:

    V(t)=K.sub.V  V.sub.x (t)R.sub.x (t)+V.sub.y (t)R.sub.y (t)!+K.sub.F  F.sub.x (t)+F.sub.y (t)!                                 (6)

where K_(V) and K_(F) are constants,

    V.sub.x (t)=V.sub.mxr (t)ω.sub.xr cos (ω.sub.xr t+ψ.sub.xr)

    V.sub.y (t)=V.sub.myr (t)ω.sub.yr cos (ω.sub.yr t+ψ.sub.yr)(7)

    R.sub.x (t)=d.sub.i cos (2θ-2θ.sub.r) cos (ωt+φ)-d.sub.q sin (2θ-2θ.sub.r) sin (ωt+φ)

    R.sub.y (t)=d.sub.i sin (2θ-2θ.sub.r) cos (ωt+φ)+d.sub.q cos (2θ-2θ.sub.r) sin (ωt+φ)

and

    F.sub.x (t)=V.sub.cx (t) ω.sub.Uxp U.sub.xp (t)C.sub.xp ω.sub.Uxn U.sub.xn (t)C.sub.xn !

    F.sub.y (t)=V.sub.cy (t) ω.sub.Uyp U.sub.yp (t)C.sub.yp ω.sub.Uyn U.sub.yn (t)C.sub.yn !                    (8)

The signals R_(x) (t) and R_(y) (t) are the desired outputs from a demultiplexing process consisting of separate operations applied to V(t) since they contain the standing wave parameters d_(i), d_(q), θ-θ_(r), ω, and φ.

Signals S_(x) (t) and S_(y) (t) containing the signals R_(x) (t) and R_(y) (t) are extracted by amplifier-demultiplexer 44 by performing the operations O_(x) on S_(x) (t) and the operations O_(y) and S_(y) (t). The operating principle of the demultiplexer portion of the amplifier-demultiplexer 44 depends on the form of the voltages V_(mxr) (t), V_(myr) (t), V_(cx) (t), and V_(cy) (t) and the values of ω_(xr), ω_(yr), ψ_(xr), and ψ_(yr).

For frequency-division multiplexing, V_(mxr) (t), V_(myr) (t), V_(cx) (t), and V_(cy) (t) are all equal to a constant, ω_(xr), ω_(yr), and |ω_(xr) -ω_(yr) | are greater than about 6ω, and ψ_(xr), and ψ_(yr) are arbitrary constants. The signals R_(x) (t) and R_(y) (t) which contain the standing-wave parameters are obtained by performing two product demodulations of V(t), one with respect to cos(ω_(xr) t+ψ_(xr)) and the other with respect to cos(ω_(yr) t+ψ_(yr)). If a periodic function other than a sinusoid is being used, then the demodulations proceed using replicas of the periodic functions. A product demodulation consists of multiplying the input voltage by the reference sinusoid (or replica) and lowpass filtering the product, the cutoff frequency of the lowpass filter being about 3ω. The results of the above process are the signals S_(FDMx) (t) and S_(FDMy) (t):

    S.sub.FDMx (t)=K.sub.FDM R.sub.x (t)

    S.sub.FDMy (t)=K.sub.FDM R.sub.y (t)                       (9)

where K_(FDM) is a constant. Because the upper limit to the frequency spectrums of F_(x) (t) and F_(y) (t) are about 3ω, these quantities are eliminated by the demultiplexing process.

For phase-division multiplexing, ω_(xr) and ω_(yr) have the same value ω_(o), ω_(o) being greater than about 6ω, and ψ_(xr) -ψ_(yr) is equal to π/2 radians. The signals S_(PDMx) (t) and S_(PDMy) (t) are obtained by performing product demodulations of V(t) with respect to cos(ω_(o) t+ψ_(x)) and with respect to cos(ω_(o) t+ψ_(y)) (or with respect to replicas of the periodic functions being used).

    S.sub.PDMx (t)=K.sub.PDM R.sub.x (t)

    S.sub.PDMy (t)=K.sub.PDM R.sub.y (t)                       (10)

where K_(PDM) is a constant.

For one form of time-division multiplexing, ω_(xr) and ω_(yr) have the same value ω_(o) with ω_(o) being greater than about 6ω and ψ_(xr), and ψ_(yr) are equal to an arbitrary number ω_(o). The voltages V_(mxr) (t) and V_(myr) (t) are proportional to square waves which take on values of 0 and 1, only one of which being equal to 1 at any given time and the duration of a "1" value being equal to an integer times 2π/ω. The voltages V_(cx) (t), and V_(cy) (t) are both equal to a constant. The signals S_(TDMx) (t) and S_(TDMy) (t) are obtained by performing a product demodulation of V(t) with respect to cos(ω_(o) t+ψ_(o)) (or replica) followed by parallel multiplications with V_(mxr) (t) and V_(myr) (t):

    S.sub.TDMx (t)=K.sub.TDM V.sub.mxr (t)R.sub.x (t)

    S.sub.TDMy (t)=K.sub.TDM V.sub.myr (t)R.sub.y (t)          (11)

where K_(TDM) is a constant. It should be noted that R_(x) (t) and R_(y) (t) are available only when V_(mxr) (t) and V_(myr) (t) are non-zero.

The same results are obtained (except possibly for the value of the constant K_(TDM)) if V_(mxr) (t), V_(myr) (t), V_(cx) (t), and V_(cy) (t) are proportional to square waves which take on values of 0 and 1, only one of the square waves being equal to 1 at any given time, and the duration of a "1" value being equal to an integer times 2π/ω. This mode of operation may be desirable in that it completely isolates the forcing voltages V_(cx) (t)U_(xp) (t), V_(cx) (t)U_(xn) (t), V_(cy) (t)U_(yp) (t), and V_(cy) (t)U_(yn) (t) from each other and from the excitation voltages V_(mxr) (t)cos(ω_(o) t+ψ_(o)) and V_(myr) (t)cos(ω_(o) t+ψ_(o)).

For another form of time-division multiplexing, ω_(o) equals 0 and V_(mxr) (t), V_(myr) (t), V_(cx) (t), and V_(cy) (t) are proportional to square waves which take on values of 0 and 1, only one of the square waves being equal to 1 at any given time, and the duration of a "1" value being equal to an integer times 2π/ω. Multiplying V(t) in parallel operations by V_(mxr) (t) and by V_(myr) (t) gives the same results as in the first form of time-division multiplexing.

For code-division multiplexing, ω_(xr), ω_(yr), ψ_(xr), and ψ_(yr) are all equal to 0, V_(cx) (t), and V_(cy) (t) are constants, and V_(mxr) (t) and V_(myr) (t) are proportional to square waves which take on pseudo-random sequences of values of -1/T and 1/T and satisfy the following conditions: ##EQU2## where the subscripts i and j stand for any of the subscripts mxr, myr, cx, and cy. The integration time interval T should be less than 2π/3ω. The signals S_(CDMx) (t) and S_(CDMy) (t) are obtained by separately multiplying V(t) by V_(mxr) (t) and V_(myr) (t) and then integrating over T:

    S.sub.CDMx (nT)=K.sub.CDM R.sub.x (nT)

    S.sub.CDMy (nT)=K.sub.CDM R.sub.y (nT)                     (13)

where K_(TDM) is a constant and n is an integer. It should be noted that the signals S_(CDMx) (t) and S_(CDMy) (t) provide information concerning the standing-wave parameters at intervals of T.

The voltages U_(x) (t) and U_(y) (t) typically may include three components:

    U.sub.xp (t)=U.sub.axp (t)+U.sub.qxp (t)+U.sub.rxp (t)

    U.sub.xn (t)=U.sub.axn (t)+U.sub.qxn (t)+U.sub.rxn (t)     (14)

    U.sub.yp (t)=U.sub.ayp (t)+U.sub.qyp (t)+U.sub.ryp (t)

    U.sub.yn (t)=U.sub.ayn (t)+U.sub.qyn (t)+U.sub.ryn (t)

where the subscripts a, q, and r identify the amplitude, quadrature and rate control voltages. It is not necessary to isolate these components from one another in all applications. However, if isolation is desired, the following substitutions can be made in the foregoing equations.

    V.sub.cax (t)U.sub.axp (t)+V.sub.cqx (t)U.sub.qxp (t)+V.sub.crx (t)U.sub.rxp (t) for V.sub.cx (t)U.sub.xp (t)

    V.sub.cax (t)U.sub.axn (t)+V.sub.cqx (t)U.sub.qxn (t)+V.sub.crx (t)U.sub.rxn (t) for V.sub.cx (t)U.sub.xn (t)

    V.sub.cay (t)U.sub.ayp (t)+V.sub.cqy (t)U.sub.qyp (t)+V.sub.cry (t)U.sub.ryp (t) for V.sub.cy (t)U.sub.yp (t)             (15)

    V.sub.cay (t)U.sub.ayn (t)+V.sub.cqy (t)U.sub.qyn (t)+V.sub.cry (t)U.sub.ryn (t) for V.sub.cy (t)U.sub.yn (t)

With these substitutions, any constraints imposed on V_(cx) (t) and V_(cy) (t) also apply to V_(cax) (t), V_(cqx) (t), V_(crx) (t), V_(cay) (t), V_(cqy) (t), and V_(cry) (t). For example, equations (1) become

    V.sub.xp (t)=V.sub.mxr (t) cos 2θ.sub.r cos (ω.sub.xr t+ψ.sub.xr)-V.sub.myr (t) sin 2θ.sub.r cos (ω.sub.yr t+ψ.sub.yr)+V.sub.cax (t)U.sub.axp (t)+V.sub.cqx (t)U.sub.qxp (t)+V.sub.crx (t)U.sub.rxp (t)

    V.sub.xn (t)=-V.sub.mxr (t) cos 2θ.sub.r cos (ω.sub.x rt+ψ.sub.xr)+V.sub.myr (t) sin 2θ.sub.r cos (ω.sub.yr t+ψ.sub.yr)+V.sub.cax (t)U.sub.axn (t)+V.sub.cqx (t)U.sub.qxn (t)+V.sub.crx (t)U.sub.rxn (t)

    V.sub.yp (t)=V.sub.mxr (t) sin 2θ.sub.r cos (ω.sub.xr t+ψ.sub.xr)+V.sub.myr (t) cos 2θ.sub.r cos (ω.sub.yr t+ψ.sub.yr)+V.sub.cay (t)U.sub.ayp (t)+V.sub.cqy (t)U.sub.qyp (t)+V.sub.cry (t)U.sub.ryp (t)                            (16)

    V.sub.yn (t)=-V.sub.mxr (t) sin 2θ.sub.r cos (ω.sub.xr t+ψ.sub.xr)-V.sub.myr (t) cos 2θ.sub.r cos (ω.sub.yr t+ψ.sub.yr)+V.sub.cay (t)U.sub.ayn (t)+V.sub.cqy (t)U.sub.qyn (t)+V.sub.cry (t)U.sub.ryn (t)

One possible time-division-multiplex configuration is a sixteen-slot frame of duration 32π/ω synchronized to the flexure rate of the resonator. The multiplex control voltages are as shown in FIG. 3. When θ_(r) equals θ, the x_(r) axes coincide with the antinodal axes and the y_(r) axes coincide with the nodal axes. Eight slots are assigned to reading out the y_(r) signal component, 4 slots to reading out the x_(r) signal component, and 1 slot each to applying amplitude, quadrature, and rate forces to the resonator. For a vibration frequency of 4 kHz, readouts of the x_(r) and y_(r) signal components would be available at a rate of 2 kHz and 1 kHz respectively. The control voltages would be applied at a rate of 0.25 kHz.

In general, the signals S_(x) (t) and S_(y) (t) exiting from the amplifier-demultiplexer 44 have the form

    S.sub.x (t)=K.sub.Vx R.sub.x (t)

    S.sub.y (t)=K.sub.Vy R.sub.y (t)                           (17)

where K_(Vx) and K_(Vy) each equals K_(V) except in the case of time-division multiplexing when K_(Vx) equals K_(V) V_(mx) (t) and K_(Vy) equals K_(V) V_(my) (t).

In order to extract the standing-wave parameters from the signals S_(x) (t) and S_(y) (t), a stable and precise replica of the resonator vibration signal cos(ωt+φ) is required. The replica is obtained from a voltage-controlled oscillator in replica generator 52 wherein the voltage-controlled oscillator is phase-locked to the in-phase standing-wave antinodal signal. The first step of the process is to multiply S_(x) (t) and S_(y) (t) first by the replica signal cos(ω_(r) t+φ_(r)) and lowpass filter the results and then by the phase-shifted replica sin(ω_(r) t+φ_(r)) and lowpass filter the results. The results of this process are: ##EQU3## where K is a constant.

The next step is to form the following combinations of products of the S_(ix) (t), S_(iy) (t), S_(qx) (t), and S_(qy) (t):

    E=S.sub.ix.sup.2 +S.sub.qx.sup.2 +S.sub.iy.sup.2 +S.sub.qy.sup.2 =K.sup.2 (d.sub.i.sup.2 +d.sub.q.sup.2)

    Q=2(S.sub.ix S.sub.qy -S.sub.iy S.sub.qx)=K.sup.2 (2d.sub.i d.sub.q)

    R=S.sub.ix.sup.2 +S.sub.qx.sup.2 -S.sub.iy.sup.2 -S.sub.qy.sup.2 =K.sup.2 (d.sub.i.sup.2 -d.sub.q.sup.2) cos (4θ-4θ.sub.r)(19)

    S=2(S.sub.ix S.sub.iy +S.sub.qx S.sub.qy)=K.sup.2 (d.sub.i.sup.2 -d.sub.q.sup.2) sin (4θ-4θ.sub.r)

    L.sub.i =2(S.sub.ix S.sub.qx +S.sub.iy S.sub.qy)=K.sup.2 (d.sub.i.sup.2 -d.sub.q.sup.2) sin  2(ω.sub.r -ω)t+2(φ.sub.r -φ)!

With L_(i) (t) as the error signal, the phase-locked loop will lock up with the replica phase φ_(r) equal to φ and ω_(r) equal to ω.

The difference between the standing-wave orientation angle and the tracking angle θ-θ_(r), can be determined from the equation ##EQU4## and the signs of S_(ix) (t) and S_(iy) (t) The quantity S(t) can be used as the error signal in a control loop which generates θ_(r) and causes on average θ to equal θ_(r) and d/dt(θ-θ_(r)) to equal 0. The digitally-synthesized tracking angle θ_(r) is used in generating sinθ_(r) and cosθ_(r) which are supplied to the multiplexer 48. The actual value of θ at any given time is given by ##EQU5##

The difference between E(t) and a specified number is used as the error signal in the amplitude control loop which causes the total energy in the combined inphase and quadrature standing waves, which is proportional to d_(i) ² +d_(q) ², to equal the specified number.

The quantity Q(t) is used as the error signal in the quadrature control loop which results in the quadrature standing-wave amplitude d_(q) to be zero. When this loop is closed, the amplitude control loop maintains the inphase amplitude d_(i) at a specified value.

The use of the above control variables can be shown to be optimum. It will be apparent to those skilled in the art that there are many choices of control variables that are suboptimum but still practical.

The outputs of the control unit 50 are the functions U_(xp) (t), U_(xn) (t), U_(yp) (t), and U_(yn) (t) together with the sine and cosine of θ_(r), which are all supplied to multiplexer 48.

Additional details concerning vibratory rotation sensors are contained in U.S. Pat. No. 4,951,508 by Loper, Jr. et al. dated Aug. 28, 1990 which is incorporated by reference. 

What is claimed is:
 1. A vibratory rotation sensor comprising:a resonator, the resonator being a rotationally-symmetric thin-walled object, the resonator being capable of vibrating in at least one of a plurality of standing-wave modes, the orientation of a standing wave with respect to a reference point on the resonator being specified by an orientation angle, one or more electrodes being attached to a surface of the resonator, the one or more electrodes being electrically connected to a single output port; sensing electronics having an input port which is connected to the output port of the resonator, the sensing electronics obtaining from a resonator signal available at the output port of the resonator a measure of the difference between the orientation angle of a standing wave and a tracking angle, the tracking angle being with respect to the reference point on the resonator.
 2. The vibratory rotation sensor of claim 1 wherein the signal sum V_(x) (t)R_(x) (t)+V_(y) (t)R_(y) (t) is among the signals present in the resonator signal, V_(x) (t) and V_(y) (t) being predetermined functions of time t and R_(x) (t) and R_(y) (t) being functions of time, the difference Δθ between the orientation angle of a standing wave and the tracking angle, and the other parameters of the one or more standing waves, the sensing electronics separately performing operations O_(x) and O_(y) on the resonator signal to obtain scaled versions of R_(x) (t) and R_(y) (t) respectively.
 3. The vibratory rotation sensor of claim 2 wherein O_(x) includes multiplying the resonator signal by a replica of a periodic function of angular frequency ω_(x) and phase ψ_(x) followed by lowpass filtering and O_(y) includes multiplying the resonator signal by a replica of a periodic function of angular frequency ω_(y) and phase ψ_(y) followed by lowpass filtering, the values of ω_(x), ψ_(x), ω_(y), and ψ_(y) being predetermined.
 4. The vibratory rotation sensor of claim 2 wherein O_(x) includes multiplying the resonator signal by a replica of a periodic function of angular frequency ω_(o) and phase ψ_(o) followed by lowpass filtering and O_(y) includes multiplying the resonator signal by a replica of the periodic function of phase (ψ_(o) +π/2) followed by lowpass filtering, the values of ω_(o) and ψ_(o) being predetermined.
 5. The vibratory rotation sensor of claim 2 wherein O_(x) includes multiplying the resonator signal by V_(mxr) (t) and O_(y) includes multiplying the resonator signal by V_(myr) (t) where V_(mxr) (t) and V_(myr) (t) are proportional to predetermined square-wave functions of time which take on values of 0 and 1, the square-wave functions not being equal to 1 at the same time.
 6. The vibratory rotation sensor of claim 2 wherein (1) O_(x) includes (a) multiplying the resonator signal by a replica of a periodic function followed by lowpass filtering and (b) multiplying by V_(mxr) (t) and (2) O_(y) includes (a) multiplying the resonator signal by the replica of the periodic function followed by lowpass filtering and (b) multiplying by V_(myr) (t), the replica of the periodic signal having an angular frequency of ω_(o) and phase ψ₀, the values of ω_(o) and ψ_(o) being predetermined, V_(mxr) (t) and V_(myr) (t) being proportional to predetermined square-wave functions of time which take on values of 0 and 1, the square-wave functions not being equal to 1 at the same time.
 7. The vibratory rotation sensor of claim 2 wherein O_(x) includes multiplying the resonator signal by V_(mxr) (t) followed by an integration over a predetermined time period T and O_(y) includes multiplying the resonator signal by V_(myr) (t) followed by an integration over time period T where V_(mxr) (t) and V_(myr) (t) are proportional to square wave functions of time which take on sequences of values -1 and 1 during time period T.
 8. The vibratory rotation sensor of claim 2 wherein R_(x) (t) and R_(y) (t) are given by the equations

    R.sub.x (t)=d.sub.i cos (2Δθ) cos (ωt+φ)-d.sub.q sin (2Δθ) sin (ωt+φ)

    R.sub.y (t)=d.sub.i sin (2Δθ) cos (ωt+φ)+d.sub.q cos (2Δθ) sin (ωt+φ)

where d_(i) and d_(q) are the flexing amplitudes respectively of the inphase and quadrature vibration modes, ω is the angular frequency of vibration of the resonator, and φ is the phase of vibration.
 9. The vibratory rotation sensor of claim 1 further comprising:a housing to which the resonator is attached, the housing having a plurality of attached electrodes in close proximity to the one or more resonator electrodes; driving electronics which supplies a voltage V_(x1) (t) to a first housing electrode and a voltage V_(y2) (t) to a second housing electrode, V_(x1) (t) including voltage components V_(mxr) (t)cos(2θ_(r))F(ω_(xr) t+ψ_(xr)) and V_(myr) (t)sin(2θ_(r))F(ω_(yr) t+ψ_(yr)) and V_(y2) (t) including voltage components V_(mxr) (t)sin(2θ_(r))F(ω_(xr) t+ψ_(xr)) and V_(myr) (t)cos(2θ_(r))F(ω_(yr) t+ψ_(yr)) where V_(mxr) (t) and V_(myr) (t) are either predetermined functions of time t or constants, θ_(r) is the tracking angle, and F(ωt+ψ) is a periodic function of time t with frequency ω and phase ψ, the values of ω_(xr), ψ_(xr), ω_(yr), and ψ_(yr) being predetermined, V_(mxr) (t)F(ω_(xr) t+ψ_(xr)) and V_(myr) (t)F(ω_(yr) t+ψ_(yr)) having no significant effect on the standing-wave dynamics of the resonator.
 10. The vibratory rotation sensor of claim 9 wherein V_(mxr) (t) and V_(myr) (t) are constants and ω_(xr), ω_(yr), and |ω_(xr) -ω_(yr) | are greater than about 6ω, ω being the angular frequency of vibration of the resonator.
 11. The vibratory rotation sensor of claim 9 wherein V_(mxr) (t) and V_(myr) (t) are constants, ω_(xr) and ω_(yr), are equal to a predetermined value ω_(o), and ψ_(xr) -ψ_(yr) is equal to π/2 radians, ω_(o) being greater than about 6ω, ω being the angular frequency of vibration of the resonator.
 12. The vibratory rotation sensor of claim 9 wherein ω_(xr), ω_(yr), ψ_(xr), and ψ_(yr) are equal to 0 and V_(mxr) (t) and V_(myr) (t) are proportional to first and second square-wave functions of time respectively which take on values of 0 and 1, the first and second square-wave functions not being equal to 1 at the same time.
 13. The vibratory rotation sensor of claim 9 wherein ω_(xr) and ω_(yr) are equal to a predetermined value ω_(o), ψ_(xr) and ψ_(yr) are equal to a predetermined value ψ_(o), and V_(mxr) (t) and V_(myr) (t) are proportional to first and second square-wave functions respectively which take on values of 0 and 1, ω_(o) being greater than about 6ω where ω is the angular frequency of vibration of the resonator, the first and second square-wave functions not being equal to 1 at the same time.
 14. The vibratory rotation sensor of claim 9 wherein ω_(xr), ω_(yr), ψ_(xr), and ψ_(yr) are equal to 0 and V_(mxr) (t) and V_(myr) (t) are proportional to first and second square-wave functions respectively which take on pseudorandom sequences of values of -1 and
 1. 15. The vibratory rotation sensor of claim 9 wherein V_(x1) (t) and V_(y2) (t) also includes voltage components V_(cx) (t)U_(x1) (t) and V_(cy) (t)U_(y2) (t) respectively, the quantities V_(cx) (t) and V_(cy) (t) being either fumctions of time t or constants, the voltage components V_(cx) (t)U_(x1) (t) and V_(cy) (t)U_(y2) (t) resulting in forces being applied to the resonator.
 16. The vibratory rotation sensor of claim 15 wherein V_(mxr) (t), V_(myr) (t), V_(cx) (t), and V_(cy) (t) are constants and ω_(xr), ω_(yr), and |ω_(xr) -ω_(yr) | are greater than about 6ω, ω being the angular frequency of vibration of the resonator.
 17. The vibratory rotation sensor of claim 15 wherein V_(mxr) (t), V_(myr) (t), V_(cx) (t), and V_(cy) (t) are constants, ω_(xr) and ω_(yr) are equal to a predetermined number ω_(o), and ψ_(xr) -ψ_(yr) is equal to π/2 radians, ω_(o) being greater than about 6ω, ω being the angular frequency of vibration of the resonator.
 18. The vibratory rotation sensor of claim 15 wherein ω_(xr), ω_(yr), ψ_(xr), and ψ_(yr) are equal to 0 and V_(mxr) (t), V_(myr) (t), V_(cx) (t), and V_(cy) (t) are proportional to first, second, third, and fourth square-wave functions of time respectively which take on values of 0 and 1, the first, second, third, and fourth square-wave functions not being equal to 1 at the same time.
 19. The vibratory rotation sensor of claim 15 wherein ω_(xr) and ω_(yr) are equal to a predetermined value ω_(o), ψ_(xr) and ψ_(yr) are equal to a predetermined value ψ_(o), and V_(mxr) (t), V_(myr) (t), V_(cx) (t), and V_(cy) (t) are proportional to first, second, third, and fourth square-wave functions of time respectively which take on values of 0 and 1, ω_(o) being greater than about 6ω where ω is the angular frequency of vibration of the resonator, the first, second, third, and fourth square-wave functions not being equal to 1 at the same time.
 20. The vibratory rotation sensor of claim 15 wherein V_(x1) (t) also includes voltage components V_(cax) (t)U_(ax1) (t), V_(cqx) (t)U_(qx1) (t), and V_(crx) (t)U_(rx1) (t) and V_(y2) (t) also includes voltage components V_(cay) (t)U_(ay2) (t), V_(cqy) (t)U_(qy2) (t), and V_(cry) (t)U_(ry2) (t), ω_(x), ω_(y), ψ_(x), and ψ_(y) being equal to 0, V_(mxr) (t), V_(myr) (t), V_(cax) (t), V_(cax) (t), V_(crx) (t), V_(cay) (t), V_(cqy) (t), and V_(cry) (t) being proportional to first, second, third, fourth, fifth, sixth, seventh, and eighth square-wave functions of time respectively which take on values of 0 and 1, the first, second, third, fourth, fifth, sixth, seventh, and eighth square-wave functions not being equal to 1 at the same time.
 21. The vibratory rotation sensor of claim 15 wherein V_(x1) (t) also includes voltage components V_(cax) (t)U_(ax1) (t), V_(cqx) (t)U_(qx1) (t), and V_(crx) (t)U_(rx1) (t) and V_(y2) (t) also includes voltage components V_(cay) (t)U_(ay2) (t), V_(cqy) (t)U_(qy2) (t), and V_(cry) (t) U_(ry2) (t), ω_(x) and ω_(y) being equal to a predetermined value ω_(o), ψ_(x) and ψ_(y) being equal to a predetermined value ψ_(o), ω_(o) being greater than about 6ω where ω is the angular frequency of vibration of the resonator, V_(mxr) (t), V_(myr) (t), V_(cax) (t), V_(cqx) (t), V_(crx) (t), V_(cay) (t), V_(cqy) (t), and V_(cry) (t) being proportional to first, second, third, fourth, fifth, sixth, seventh, and eighth square-wave functions of time respectively which take on values of 0 and 1, the first, second, third, fourth, fifth, sixth, seventh, and eighth square-wave functions not being equal to 1 at the same time.
 22. The vibratory rotation sensor of claim 15 wherein ω_(xr), ω_(yr), ψ_(xr), and ω_(yr) are equal to 0 and V_(mxr) (t) and V_(myr) (t) are proportional to first and second square-wave functions of time respectively which take on pseudorandom sequences of values of -1 and 1, V_(cx) (t) and V_(cy) (t) being constants.
 23. The vibratory rotation sensor of claim 1 further comprising:a housing to which the resonator is attached, the housing having a plurality of attached electrodes in close proximity to the one or more resonator electrodes; driving electronics which supplies control voltages to one or more housing electrodes, the driving electronics generating the tracking angle, the driving electronics causing the tracking angle to equal a constant or a function of one or more of a plurality of variables, the plurality of variables including time, the orientation of the standing wave, and variables obtained from external sources.
 24. The vibratory rotation sensor of claim 1 further comprising:a housing to which the resonator is attached, the housing having a plurality of attached electrodes in close proximity to the one or more resonator electrodes; driving electronics which supplies control voltages to one or more housing electrodes, the driving electronics generating the tracking angle, the driving electronics determining the orientation angle of the standing wave by adding the tracking angle to the difference between the orientation angle and the tracking angle.
 25. A method for controlling and reading out a vibratory rotation sensor comprising a resonator having one or more electrodes connected to a single output port and a housing having a plurality of electrodes in close proximity to the resonator electrode(s), the resonator being capable of vibrating in one or more standing-wave modes, each standing wave mode being defined by a plurality of parameters, the orientation of a standing wave with respect to a reference point on the resonator being specified by an orientation angle, the method comprising the steps:(a) specifying a tracking angle with respect to the reference point on the resonator; (b) generating a plurality of driving voltages which are functions of the tracking angle; (c) applying a driving voltage to each of a plurality of housing electrodes; (d) determining the difference between the orientation angle of a standing wave and the tracking angle by performing operations on a resonator signal appearing at the output port of the resonator.
 26. The method of claim 25 wherein each driving voltage includes a first excitation voltage and a second excitation voltage, the frequency spectrums of the first and second excitation voltages being confined to separated frequency bands.
 27. The method of claim 25 wherein the number of different driving voltages is two, each driving voltage including a first excitation voltage and a second excitation voltage, each excitation voltage being a periodic function of time with a predetermined frequency and a predetermined phase, the frequencies of the first and second excitation voltages being the same, the phases differing by a quarter of a cycle.
 28. The method of claim 25 wherein each driving voltage includes a first excitation voltage and a second excitation voltage, each of the first and second excitation voltages being proportional to a unique square wave that takes on the values 0 and 1, only one of the square waves taking on the value 1 at any given time.
 29. The method of claim 25 wherein each driving voltage includes a first excitation voltage and a second excitation voltage, each of the first and second excitation voltages being proportional to the product of a periodic function having a predetermined frequency and phase and a unique square wave that takes on the values 0 and 1, only one of the square waves taking on the value 1 at any given time.
 30. The method of claim 25 wherein each driving voltage includes a first excitation voltage and a second excitation voltage, each of the first and second excitation voltages being proportional to a unique square wave which takes on the values of -1 and 1 in accordance with a predetermined pseudorandom sequence.
 31. The method of claim 25 wherein each driving voltage includes a first excitation voltage and a second excitation voltage, the first excitation voltage including a multiplicative factor equal to the cosine of twice the tracking angle, the second excitation voltage including a multiplicative factor equal to the sine of twice the tracking angle.
 32. The method of claim 25 wherein each of at least two driving voltages includes a first excitation voltage, a second excitation voltage, and a forcing voltage, the frequency spectrums of the first and second excitation voltages being confined to separated frequency bands, the frequency spectrum of the forcing voltages being confined to a frequency band separated from the frequency bands associated with the first and second excitation voltages.
 33. The method of claim 25 wherein the number of different driving voltages is two, each driving voltage including a first excitation voltage, a second excitation voltage, and a forcing voltage, each of the first and second excitation voltages being a periodic function with a predetermined frequency and a predetermined phase, the frequencies of the first and second excitation voltages being the same, the phases differing by a quarter of a cycle, the frequency spectrum of the forcing voltages being confined to a frequency band separated from the frequency of the first and second excitation voltages.
 34. The method of claim 25 wherein each of at least two driving voltages includes a first excitation voltage, a second excitation voltage, and a forcing voltage, each of the first and second excitation voltages being proportional to a unique square wave that takes on the values 0 and 1, each forcing voltage including a multiplicative factor proportional to a square wave that takes on the values 0 and 1, only one of the square waves associated with the excitation and forcing voltages taking on the value 1 at any given time.
 35. The method of claim 25 wherein each of at least two driving voltages includes a first excitation voltage, a second excitation voltage, and a forcing voltage, each of the first and second excitation voltages being proportional to the product of a periodic function with a predetermined frequency and a predetermined phase and a unique square wave that takes on the values 0 and 1, each forcing voltage including a multiplicative factor proportional to a square wave that takes on the values 0 and 1, only one of the square waves associated with the excitation and forcing voltages taking on the value 1 at any given time.
 36. The method of claim 25 wherein each of at least two driving voltages includes a first excitation voltage, a second excitation voltage, and a forcing voltage, each of the first and second excitation voltages being proportional to a unique square wave which takes on the values of -1 and 1 in accordance with a predetermined pseudorandom sequence, the frequency spectrum of the forcing voltages being confined to a frequency band separated from the frequency band associated with the excitation voltages.
 37. The method of claim 25 wherein the resonator signal is the sum of two components that are functions of the parameters of the standing waves and the tracking angle, step (d) comprising the steps:(d1) extracting each of the two components from the resonator signal; (d2) determining the difference between the orientation angle of one of the standing-waves and the tracking angle by performing operations on the two components.
 38. The method of claim 37 wherein the two components occupy separated frequency bands, each component being extracted by performing operations on the resonator signal that discriminate between the separated frequency bands.
 39. The method of claim 37 wherein the two components are periodic functions of time having the same frequency and phases that differ by one-quarter of a cycle, each component being extracted by performing operations on the resonator signal that discriminate between the phases of the two components.
 40. The method of claim 37 wherein the two components are present in the resonator signal during different time periods, each component being extracted by performing operations on the resonator signal that discriminate between the different time periods.
 41. The method of claim 37 wherein the two components are pseudorandom sequences of 0's and 1's, the cross correlation of the two pseudorandom sequences being equal to 0, each component being extracted by performing operations on the resonator signal that discriminate between the two pseudorandom sequences.
 42. The method of claim 37 wherein each of the two components a sum of two terms, one term containing the sine of the difference between the orientation angle and the tracking angle, the other term containing the cosine of the difference between the orientation angle and the tracking angle.
 43. The method of claim 37 further comprising the step:(e) causing the tracking angle to equal a constant or a function of one or more of a plurality of variables, the plurality of variables including time, the orientation of the standing wave, and variables obtained from external sources.
 44. The method of claim 43 further comprising the step:(f) calculating orientation angle by adding the tracking angle to the measured difference between the orientation angle and the tracking angle. 